Local and global cyclicity in free semigroups

Holub, Štěpán

doi:10.1016/s0304-3975(00)00156-0

Cited by 14 publications

(36 citation statements)

References 6 publications

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“…Definition 5.1. A nonempty word w is a solution to (8) if w can be written as a product of minimal square roots w = X 1 X 2 · · · X n which satisfy the word equation (8). The solution is trivial if…”

Section: Characterization By a Word Equationmentioning

confidence: 99%

“…All minimal square roots of slope α are trivial solutions to (8). One example of a nontrivial solution is w = S 2 S 1 S 4 in the language of the Fibonacci word (i.e., in the language of slope [0; 2, 1, 1, .…”

Section: Characterization By a Word Equationmentioning

confidence: 99%

“…Lemma 5.6. Let w be a primitive solution to (8) having the word S 6 = 10 a+1 (10 a ) b+1 as a suffix such that w 2 , L(w) ∈ L(a, b). Then wL(w) ∈ Π(a, b) and wL(w) = w.…”

Section: Lemma 55 Let U and V Be Words Such Thatmentioning

confidence: 99%

“…. , S 5 are solutions to (8), so we are left with the case where w = s 2,ℓ = 0(10 a ) ℓ for some ℓ such that 1 < ℓ ≤ b + 1. It is straightforward to see that if ℓ is even, then…”

Section: Lemma 55 Let U and V Be Words Such Thatmentioning

confidence: 99%

“…It was known that the word equation (X 2 1 · · · X 2 n ) = X 2 1 · · · X 2 n has nonperiodic solutions [7], but according to our knowledge no large families of nonperiodic solutions have been identified until our result. Word equations of the type X k 1 · · · X k n = (X 1 · · · X n ) k have been considered by Štěpán Holub [6,7,8]. The final central topic of this paper concerns the square root map in a more general setting.…”

Section: Introductionmentioning

confidence: 99%

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A Square Root Map on Sturmian Words

Peltomäki

Whiteland

2015

Lecture Notes in Computer Science

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We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope α, there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper prefix). A Sturmian word s of slope α can be written as a product of these six minimal squares:The square root of s is defined to be the wordThe main result of this paper is that that √ s is also a Sturmian word of slope α. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of √ s and an occurrence of any prefix of √ s in s. Related to the square root map, we characterize the solutions of the word equation X 2 1 X 2 2 · · · X 2 n = (X 1 X 2 · · · X n ) 2 in the language of Sturmian words of slope α where the words X 2 i are minimal squares of slope α. We also study the square root map in a more general setting. We explicitly construct an infinite set of non-Sturmian fixed points of the square root map. We show that the subshifts Ω generated by these words have a curious property: for all w ∈ Ω either √ w ∈ Ω or √ w is periodic. In particular, the square root map can map an aperiodic word to a periodic word.

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Section: Characterization By a Word Equationmentioning

confidence: 99%

Section: Characterization By a Word Equationmentioning

confidence: 99%

“…Lemma 5.6. Let w be a primitive solution to (8) having the word S 6 = 10 a+1 (10 a ) b+1 as a suffix such that w 2 , L(w) ∈ L(a, b). Then wL(w) ∈ Π(a, b) and wL(w) = w.…”

Section: Lemma 55 Let U and V Be Words Such Thatmentioning

confidence: 99%

Section: Lemma 55 Let U and V Be Words Such Thatmentioning

confidence: 99%